Showing papers on "Graph (abstract data type) published in 1974"

Structural controllability

Abstract: The new concepts of "structure" and "structural controllability" for a linear time-invariant control system (described by a pair ( A,b )) are defined and studied. The physical justification of these concepts and examples are also given. The graph of a pair ( A,b ) is also defined. This gives another way of describing the structure of this pair. The property of structural controllability is reduced to a property of the graph of the pair ( A,b ). To do this, the basic concept of a "cactus" and the related concept of a "precactus" are introduced. The main result of this paper states that the pair ( A,b ) is structurally controllable if an only if the graph of ( A,b ) is "spanned by a cactus." The result is also expressed in a more conventional way, in terms of some properties of the pair ( A,b ).

Transformation of Multisalesman Problem to the Standard Traveling Salesman Problem

Abstract: It is shown that the multisalesmen problem can be solved by solving the standard traveling salesman problem on an expanded graph. The expanded graph has m — 1 more nodes than the original graph where m is the number of salesmen available at the base.

Some Applications of Graph Theory to Clustering.

Abstract: This paper attempts to review and expand upon the relationship between graph theory and the clustering of a set of objects. Several graphtheoretic criteria are proposed for use within a general clustering paradigm as a means of developing procedures “in between” the extremes of complete-link and single-link hierarchical partitioning; these same ideas are then extended to include the more general problem of constructing subsets of objects with overlap. Finally, a number of related topics are surveyed within the general context of reinterpreting and justifying methods of clustering either through standard concepts in graph theory or their simple extensions.

A Graph Theoretic Analysis of Pattern Classification via Tamura's Fuzzy Relation

Abstract: Graph theoretical arguments are used to show that the hierarchical clustering scheme induced by Tamura's N-step fuzzy relation f is contained in the maximal single linkage hierarchy. A method of computing f is proposed, based upon Prim's algorithm for generating maximal spanning trees and a result reported by Hu on maximal capacity routes in maximal spanning trees. It is shown that this procedure is superior to Tamura's generalized matrix multiplication algorithm with regard to both computing time and storage requirements.

Efficient algorithm for the partitioning of trees

Abstract: This paper describes an algorithm for partitioning a graph that is in the form of a tree. The algorithm has a growth in computation time and storage requirements that is directly proportional to the number of nodes in the tree. Several applications of the algorithm are briefly described. In particular it is shown that the tree partitioning problem frequently arises in the allocation of computer information to blocks of storage. Also, a heuristic method of partitioning a general graph based on this algorithm is suggested.

A Search Procedure for Hamilton Paths and Circuits

Abstract: A search procedure is given which will determine whether Hamilton paths or circuits exist in a given graph, and will find one or all of them. A combined procedure is given for both directed and undirected graphs. The search consists of creating partial paths and making deductions which determine whether each partial path is a section of any Hamilton path whatever, and which direct the extension of the partial paths.

Cost-minimal trees in directed acyclic graphs

Abstract: A number of examples are given where one seeks a minimal-cost tree to span a given subset of the nodes of a connected, directed, acyclic graph (we call such a graph monotonic). Some of these examples require an algorithm to transform the problem into the form of a minimization problem in a monotonic graph; this algorithm is also given. Finally, an implicit enumeration algorithm is presented for finding the cost-minimal tree of the graph, which spans the designated subset of the nodes, and some computational results are given.

The bordered triangular matrix and minimum essential sets of a digraph

Abstract: A partitioning strategy of sparse matrices is dealt with. In particular, the problem of transforming a nonsingular matrix by symmetric permutation to an optimal bordered triangular form (BTF) is solved. It is shown that the problem is equivalent to the determination of a minimum essential set of a directed graph. An efficient algorithm is given for finding minimum essential sets of a digraph. The method depends on, as a preliminary step, graph simplication using local information at a vertex. A circuit-generation technique based on vertex elimination is then introduced. The algorithm is illustrated with a complete example. A simple electrical network is used to illustrate the use of the BTF in the sparse tableau approach of network analysis.

Communication networks: Message path delays

Abstract: A communication network is modeled by a weighted graph. The vertices of the graph represent stations with storage capabilities, while the edges of the graph represent communication channels (or other information processing media). Channel capacity weights are assigned to the edges of the network. The network is assumed to operate in a store-and-forward manner, so that when a channel is busy the messages directed into it are stored at the station, where it joins a queue that is governed by a first-come first-served service discipline. Assuming that fixed-length messages arrive at random at the network, following the statistics of a Poisson point process, we solve for the steady-state distributions of the message overall delay time, for the average message waiting times at the individual stations, for the average memory size requirements at the stations, as well as for other statistical characteristics of the message flow along a communication path.

General passive networks-Solvability, degeneracies, and order of complexity

Abstract: A topological expansion of the network determinant is developed which makes possible the investigation of some properties of general passive networks modeled by two-terminal elements and gyrators. Necessary and sufficient conditions are obtained for the unique solvability of the network equations. These conditions are expressed in terms of the network graph and certain gyrator-only networks, called G -networks, which are derived from the original network by removing and contracting two-terminal elements. The solvability of G -networks is also shown to be fully related to the degeneracies existing in the state-variable characterization of the network. A procedure is given for determining the actual number of state equations (order of complexity) needed for describing an arbitrarily degenerate passive network. The theoretical results obtained are illustrated with some examples.

Is the null-graph a pointless concept?

Abstract: The graph with no points and no lines is discussed critically. Arguments for and against its official admittance as a graph are presented. This is accompanied by an extensive survey of the literature. Paradoxical properties of the null-graph are noted. No conclusion is reached.

Structures of influence and cooperation‐conflict

Abstract: This paper outlines a structural approach to the analysis of patterns of influence and amity‐enmity in international politics. Structure is defined as a set of relations between pairs of social units, which justifies the use of graph theory to represent structures. Graph theoretical models of international structure are explored—one for each of the two types of relations. The author devises some typologies and observes changing structures. A tendency toward hierarchy in influence structures and toward bipolarization in cooperation‐conflict structures is discussed. It is hypothesized that high degrees of hierarchy and polarization are inimical to international peace. The graph theoretical studies yield intuitively reasonable results and the models appear to be useful in making several concepts of international relations theory more precise.

Graph Representation and Computation Rules for Typeless Recursive Languages

Abstract: Computations are carried out in recursive languages by the iterative application of two kinds of transformation rules (substitution and simplification rules) to an initial term.

On the Number of Nonzeros Added when Gaussian Elimination is Performed on Sparse Random Matrices

Abstract: This paper studies the fill-in properties of Gaussian elimination on sparse random matrices. A theoretical study using the concept of random graphs yields formulae from which the fill-in can be evaluated. The predictions are examined for matrices of various orders and densities and are found to be in close agreement with experimental results. A possible consequence of the results of this paper relating to error analysis for sparse systems is given in the concluding section. 1. Introduction. This paper studies the fill-in properties of Gaussian elimina- tion on sparse random matrices. Graph theoretical ideas are introduced and the problem is examined using the concept of random graphs. This concept of using random graphs is not a new one. They are used as a tool in the development of existence theorems in graph theory (Erdos (1967)) and some work has been done on the structure of a graph where the number of lines is a function of the number of points, the analysis holding in the limit as the number of points tends to infinity (Palasti (1966, 1970)). Ogilvie (1968) has shown that these asymptotic results hold where the graph has only a small number of vertices (about 10) but none of this work is directly concerned with matrix decomposition or indeed with the adjacency matrix of the graph. Heap (1966) has extended the notion of random graphs to examine properties of their adjacency matrices. However, his Tesults on matrix reducibility are only valid for comparatively dense matrices (prob- ability density > 0.15). A serious attempt to tackle the problem of this paper has been made by Hsieh and Ghausi (1971) where they have tried to make theoretical predictions on the results of Brayton et al. (1970). Their work has, however, several serious defects. Although they make more allowance for interaction than the discus- sion in Section 2, their formulae only include correlation between elements in the same column ignoring the correlation which exists between elements in the same row. In addition, some of their formulae break down if the density of the matrix increases above 20% and their correlation factor H is chosen empirically to obtain a fit with Brayton's data rather than determined theoretically. The impetus for writing this paper came from remarks by Robert Brayton in the panel at the end of the IBM symposium (Rose and Willoughby (1972)) when he stated that the above problem, although itself not perhaps of devastating importance, has consumed vast

A Graph-theoretical Analysis of the HOMO-LUMO Separation in Conjugated Hydrocarbons

Abstract: Graph-theoretical rules concerning the HOMO-LUMO separation in conjugated hydrocarbons are given. It has been found that a closure of odd-membered rings will not essentially change the HOMO-LUMO energy gap (ΔΕ). However, ΔΕ will be shifted down or up depending whether a (4m + 1)-or (4m + 3)-ring is closed. A closure of an even-membered ring increase the value of ΔΕ in the case of (4m + 2)- and decrease ΔΕ value in the case of (4m)-cycles if delocalized rings are formed. The effect is opposite if quinonoid rings are generated.

An indirect method for minimal essential sets

Abstract: Further results on minimal essential sets in directed graphs are presented and an indirect algorithm based on the analysis of the complementary graph is derived.

Graph isomorphism: A heuristic edge-partitioning-oriented algorithm

Abstract: A new algorithm for testing isomorphism on directed and undirected graphs is described. It consists of an 0 (n 5) algorithm for partitioning node and edge sets, plus a heuristic procedure for deriving all the isomorphisms. The partitioning is described by a “connectivity graph” on which a sufficient condition for isomerphism can be tested.

A Prototype Computer Program for Interactive Computer-Administered Admissible Probability Measurement

Abstract: : This report describes an experimental computer implementation of admissible probability measurement techniques These measurement techniques rely on a scoring system that encourages the user to reproduce or reveal his own estimate of the probability of correctness of an answer to a multiple-choice question A track record (in the form of an external validity graph) of the user's probability estimating performance provides the means for estimating how much information he has and how realistically that level of knowledge is assessed The computer system allows the user to explore the relationships between his assignment of probability of correctness to answers and the possible score outcome

Fault Diagnosis as a Graph Coloring Problem

Abstract: A method is presented for converting problems in fault diagnosis into ones involving coloring (labeling) the nodes of a graph. It is shown that many concepts and results of graph theory have direct counterparts in fault diagnosis. Specific fault diagnosis problems examined include exercising tests, fanout free networks, test point insertion, sensitized paths, and fault test generation. A number of examples are included.